As explained for example in [1]}, any tilting sequence \(\mathcal {S}=(E_1,E_2,E_3)\)  in the derived category \(\mathcal {C}=D^b\operatorname{Coh}(K_{{P}^2})\)  (in the sense that \(\mathcal {S}\)  generates \(\mathcal {C}\)  and \(\operatorname{Ext}^k(E_i,E_j)=0\)  for \(k\ne 0\)  and all \(i,j\) ) provides an isomorphism \(\mathcal {C}\sim D^b(\operatorname{Rep}J(Q,W))\) 
with the derived category of representations of the Jacobian algebra \(J(Q,W)\)  of a certain
quiver with potential \((Q,W)\)  associated to \(\mathcal {S}\) . A tilting sequence can be obtained
from any strong exceptional collection \((F_1,F_2,F_3)\)  on \({P}^2\)  by setting
\(\mathcal {S}=(i_*(F_1[1]), i_*(F_2), i_*(F_3)[-1])\) .
Starting from the strong exceptional collection \((\mathcal {O}(-1),\Omega (1),\mathcal {O})\) , one
arrives at the tilting sequence in (REF ), associated to the
quiver with potential depicted in Figure REF . All other strong exceptional collections are obtained by successive mutations of \((\mathcal {O}(-1),\Omega (1),\Omega )\) ,
and similarly lead to 3-node quivers \((Q_{a,b,c},W_{a,b,c})\)  with \(a\)  arrows
from node 1
to node 2, \(b\)  arrows from 2 to 3 and \(c\)  arrows from 3 to 1, with
\((a,b,c)\)  any set of positive integers satisfying the Markov condition \(a^2+b^2+c^2=abc\) .
